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This is the second post on a series on logic as formulated in the language of type theory.

Identity types, some of the cornerstones of the new revolution in type theory and logic, are surprisingly simple, and surprisingly complex!

Definitional equality and path equality

Unlike in usual set theory, in type theory, equality is of two forms. The first form is definitional equality, which is a judgement, and the other is a binary relation \(a = b\), which is simply a type dependent over two parameters. The two kinds of equalities are different in nature as one belongs to the metalanguage, while the other to the language itself.

Judgemental equality asserts -- as a judgement -- that two terms are definitionally the same. For instance \(1 \equiv S\ 0\). Likewise, if one follows the definition of addition, one can always get \(3 + 5 \equiv 8\) definitionally, that is, purely in the metalanguage. However, although the addition of natural numbers is commutative, it is impossible to definitionally get \(n + m \equiv m + n\) for variable terms \(n\) and \(m\).

On the other hand, path equality -- whose naming will be clear when the path structure of types is discussed -- is a type which may or may not be inhabited. For instance, there is a way to prove that \(\forall x, y : \mathbb N, x + y = y + x\) by finding a term of type \(\prod_{x, y : \mathbb N} x + y = y + x\) (by recursion).

Path induction

As the identity type \(a =_ A b\) (given, \(a, b : A, A : \mathcal U\)) can be seen as a binary relation over \(A\), the best way to define is to say that it is completely generated by reflexivity; that is, the only generator is: \(\text{refl}_ a : a =_ A a\) (for every type \(A\) and term \(a : A\)). In particular, this means that \(a = b\) for \(a \not \equiv b\) is an uninhabited type (no proofs of it).

As an initial definition of the identity type, it is in fact equivalent to definitional equality; however, the statement of such a theorem is somewhat delicate as it crosses levels in the hierarchy of languages connecting definitional equality from the metalanguage with identity types from the language itself. By this, equality has been internalized from the metalanguage to the language itself allowing us to develop on it as a proposition (or type) instead of a judgement which cannot admit argumentation.

The 'induction' over the identity type is analogous to induction on the natural numbers. In particular, \(a =_A b\) is generated by \(\text{refl}_ a : a =_ A a\) giving an introduction rule \(=_ \mathcal I \) of the form \(\Gamma, A : \mathcal U, a : A \vdash \text{refl}_ A : a =_ A a\). There is also the elimination rule, by which functions out of the identity type are constructed. Given the single introduction rule, there is a single elimination rule whereby any element of the type \(a = b\) is assumed to be the element \(\text{refl}_ a\) of type \(a =_ A a\). So, any function out of \(a = b\) is completely 'generated' by its value at \(\text{refl}_ a\). Examples below should hopefully make this clearer.

Theorems / functions over identity types

Symmetry (inverses)

Symmetry is a function from \(x = y\) to \(y = x\), in particular, \(\text{sym} : \prod_{A : \mathcal U} \prod_{x, y : A} (x = y \to y = x)\). To build this function, we only need to consider the case of the point \(\text{refl}_ x : x = x\). For this case, \(\text{sym}(A, x, x, \text{refl}_ x) = \text{refl}_ x\). We also write \(\text{sym}(A, x, y, p) = p^{-1}\) (with implicit types) because it acts as the inverse of \(p\) in the sense we will talk about in a bit.

Transitivity (composition)

This is also a function, this time an operation of 2 paths, concatenation or composition.

What we need is a function \(\cdot : \prod_{A : \mathcal U}\prod_{x, y, z : A} (x = y \times y = z) \to x = z\). By currying, that is \(A \times B \to C \cong A \to (B \to C)\), it is enough to do 2 inductions, one on \(x = y\) and one on \(y = z\).

By assuming \(x \equiv y \equiv z\) and \( p \equiv q \equiv \text{refl}_ x : x = x\) in the expression \(p \cdot q\) (type arguments implicit), we get to define \(\text{refl}_ x \cdot \text{refl}_ x :\equiv \text{refl}_ x\), and by induction thus totally defining the concatenation operation \(\cdot\) which is also a proof of the transitivity of equality.

However, one induction could have sufficed with \(x \equiv y\) and \(p \equiv \text{refl}_ x\), we can get \(\text{refl}_ x \cdot q :\equiv q\). Induction on \(y = z\) alone also works giving \(p \cdot \text{refl}_ x :\equiv p\). Since all three definitions match on the only generator of the identity type (namely, \(\text{refl}\)), they are all equivalent definitions.

Proof of inverses

A proof that the inverses defined below are true inverses consists of a function \(\text{inv} : \prod_{A:\mathcal U} \prod_{x, y : A} \prod_{p : x = y} p \cdot p^{-1} = \text{refl}_ x\). This again is done by induction on the path-type with \(x \equiv y\) and \(p = \text{refl}_ x\). Then,

\(\begin{align} p \cdot p^{-1} &\equiv \text{refl}_ x \cdot (\text{refl}_ x)^{-1} \ &\equiv \text{refl}_ x \cdot \text{refl}_ x \ &\equiv \text{refl}_ x \ \end{align}\)

Since \(p \cdot p^{-1} \equiv \text{refl}_ x\), we have the term \(\text{refl}_ {\text{refl}_ x} : p \cdot p^{-1} = \text{refl}_ x\). This means that \(\text{inv}(A, x, x, \text{refl}_ x) = \text{refl}_ {\text{refl}_ x}\) thus generating all of of \(\text{inv}\) by path induction.

Groupoid structure

The above theorems (inverses, composition, identity) allow us to conclude that identity types have a groupoid structure. Groupoid means group-like, since composition is not within the same type, but rather is "transitive", more like function composition than a group operation. I will be talking about this very important structure, and extending it to the \(\infty\)-groupoid structure which represents the bigger picture. However, this not the full picture yet, as there still is the important analogy of paths in homotopy spaces, which I will also be discussing later.